SUPERSYMMETRY

A good starting point : A Supersymmetry Primer, Stephen P. Martin, 102 p.
or : Beyond the Standard Model, Dmitri Kazakov, 80 p.

A little bibliography concerning Supersymmetry is available here.

• Basics

Supersymmetry is an attractive extension of Standard Model which could provide a solution to the hierarchy problem. This additionnal symmetry preserves the masses of fundamental scalar fields, avoiding the quadratic divergences that appear in SM.

• Particle spectrum

Each matter fermion has a scalar"susy" partner. For example, there are 2 selectrons, one associated to left-handed electron, the other one to right-handed electron.

=> we talk about "left squark" although squarks have spin 0 !

On the contrary to SM, two Higgs doublets are necessary (else triangular anomalies appear). These 2 doublets describe 2 neutral complex fields and 2 charged ones (one + and one -).

Gauge and Higgs bosons have fermionic partners :

=> the photino, the Zino, two neutral Higgsinos
These 4 particles are not mass eigenstates and combine
to form 4 physical states, called the neutralinos, and
usually labelled \chi^0_i (i=1..4)
=> the winos and 2 charged higgsinos (one + and one -)
These combine to form two charginos \chi^+/-_1,2.

• Supersymmetry breaking

Supersymmetry can not be an exact symmetry of nature (else squarks and quarks would have the same mass !) Hence it has to be broken. A spontaneous breaking of susy does not lead to an acceptable spectrum. The way Susy is broken is not known.

• MSSM and its main parameters

To simplify, let's say that in MSSM (Minimal Supersymmetric Standard Model), SUSY is broken "by hand", by adding to the Lagrangian terms that are NOT supersymmetric. Of course, if you put ANY non-susy term in the Lagrangian, you will return to your starting point... i.e. quadratic divergences will come back for the Higgs mass. Therefore, the terms you can add should not break susy too strongly, in the sense that they should not bring quadratic divergences. These terms are called "soft terms of susy breaking". Such terms have been classified and you are left with quite few possibilities. Basically, you add to the supersymmetric Lagrangian :

• mass terms for the sfermions. This way, the masses of sfermions receive contributions that fermions do not, so that sfermions are heavier than leptons, precisely what you wanted !
• mass terms for the gauginos. The mass parameter for U(1) gaugino is usually called M_1, M_2 and M_3 corresponding to SU(2) and SU(3) gauginos.
• also trilinear and bilinear terms that can be important but I do not detail further.

MSSM parameters that you often find in exp. papers :

• the soft terms M_1, M_2 and M_3. Note that relations (coming from unification hypothesis at very high scale) can exist between these parameters.
• tan(beta) which is defined as the ratio of the two vacuum values of the 2 neutral Higgses
• a parameter \mu, which has the dimension of a mass, corresponding to a mass term mixing the 2 Higgses doublets. Note that \mu can be positive or negative.

Assuming the relation mentionned above between M_1 and M_2, the masses of neutralinos, charginos, and all the couplings involving neutralinos, charginos, sfermions and Standard Model fermions depend only on M_2, mu and tan(beta). Hence, experimental limits are often given in this parameter space. Note that M_3 is mainly relevant for gluino mass.

• R-parity

R-parity (Rp) is defined for each particle as Rp = (-1)^(3B+L+2S), where B, L and S are the baryonic number, the leptonic number and the spin of each particle. For all SM particles, Rp = +1, while Rp = -1 for SUSY particles. In the MSSM, Rp is assumed to be conserved multiplicatively (this ensures baryon and lepton number conservation). This has important phenomenological consequences :

• SUSY particles are always produced by pairs
• the LSP (Lightest Supersymmetric Particle) is stable (the LSP is usually assumed to be the lightest neutralino)

However, there is no a priori good reason to impose Rp conservation. The most general Lagrangian, invariant under supersymmetry and SM gauge symmetries, actually contains three kind of terms that violate Rp :


/ lepton              / lepton             / quark
slepton  /             squark  /            squark  /
---------              --------            ---------
\                     \                    \
\ lepton              \ quark              \ quark


The two first terms violate L, the last one violates B. The Yukawa couplings at the above vertices are usually labelled, respectively, lambda_ijk, lambda'_ijk, lambda''_ijk, i,j,k being generation indices.

At HERA, in case one coupling lambda'_1jk is non vanishing, it is possible to form resonantly a squark by a fusion between the incident electron/positron and a quark coming from the proton. The produced squark can then decay either via the Rp-violating coupling lambda' (leading to a process eq -> squark -> eq similar to Leptoquarks), either undergo a "gauge" decay into quark + neutralino or quark + chargino.

A key point of R-parity violating Susy models is that the LSP is now allowed to decay. For example, with a lambda' coupling :

                     / quark
/
chi^0     /
-----------
\ off-shell
\ squark
\
\---------- quark
\
\
\ lepton


  The RpV terms in the superpotential which are relevant
for HERA are :

\lambda'_{ijk} L_i Q_j D^c_k

L, Q, D^c are "superfields". Roughly, these are multiplets
with respectively :
L_i   : e^i_L  \nu^i_L  \tilde{e}^i_L   \tilde{\nu}^i_L
Q_j   : u^j_L  d^j_L    \tilde{u}^j_L   \tilde{d}^j_L
D^c_k : charge conjugates of d^k_R and \tilde{d}^k_R

To find the corresponding vertices :
you pick-up a susy particle in one
of the superfields, and one standard particle in each
remaining superfield. Use charge conservation to keep
only allowed vertices !

A coupling \lambda'_{1jk} allows the following processes :

e+ + d^k         ->  \tilde{u}^j_L
e+ + \bar{u^j}   ->  (\tilde{d}^k_R)^*

e- + u^j         -> \tilde{d}^k_R
e- + \bar{d}^k   -> (\tilde{u}^j_L)^*



• The Higgs sector

As in other models with 2 Higgs doublets :

• the real parts of the 2 (complex) neutral Higgses combine to form two neutral fields with CP = +1, labelled h and H.
• the imaginary parts of the 2 neutral Higgses combine to form two neutral fields with CP = -1, G and A.
• the 2 complex charged Higgses give charged fields H^{\pm} and G^{\pm}.
G and G^{\pm} are 3 Goldstone bosons which give their mass to the longitudinal components of electroweak gauge bosons. Finally, we are left with 5 scalar degrees of freedom :
• 2 neutral Higgses h and H, CP=+1
• 1 neutral Higgs A, CP=-1 (often called pseudo-scalar)
• 2 charged Higgses

• A last word about Constrained MSSM

The number of free parameters of the MSSM can be reduced by making the following assumptions :

• unification of the 3 gauge coupling constants at a GUT scale
• soft-breaking mass terms for the scalars at the GUT scale are the same for all scalars (this common soft breaking term is usually labelled m_0)
• also universallity of the soft-breaking mass terms for the gauginos (i.e. M_1 = M_2 = M_3 = (def) m_{1/2) at GUST scale)
• a common soft-breaking trilinear coupling A_0 at GUT scale

The only parameters of such models are then : tan(beta), m_0, m_{1/2}, A_0 and the sign of mu. Because such universality for the soft terms can be achived in local supersymmetry (also called supergravity), this model is sometimes called mSUGRA (minimal SUGRA).

In programs like SPYTHIA, SUSYGEN ... the user can give the values of the above parameters as input. Renormalization Group Equations are then solved to get the masses of all Susy particles. But you can also work in the frame of "unconstrained MSSM" and define as input the sleptons and squarks masses for example.